\[
e^{\boldsymbol{A}} = \boldsymbol{I} + \frac{\boldsymbol{A}}{1!} + \frac{\boldsymbol{A}^2}{2!} + \cdots + \frac{\boldsymbol{A}^k}{k!} + \cdots = \sum_{k=0}^{\infty} \frac{\boldsymbol{A}^k}{k!}
\]
一个关于时间的矩阵指数函数定义为
\[
e^{\boldsymbol{A}t} = \sum_{k=0}^{\infty} \frac{\boldsymbol{A}^k t^k}{k!}
\]
若矩阵指数函数对时间微分,则有
\[
\frac{\mathrm{d}}{\mathrm{d}t}(e^{\boldsymbol{A}t}) = \boldsymbol{A}e^{\boldsymbol{A}t} = e^{\boldsymbol{A}t}\boldsymbol{A}
\]
3) 矩阵乘积的微分
如果矩阵 \(\boldsymbol{A}(t)\) 和 \(\boldsymbol{B}(t)\) 对 \(t\) 是可微的,则有
\[
\frac{\mathrm{d}}{\mathrm{d}t}[\boldsymbol{A}(t)\boldsymbol{B}(t)] = \frac{\mathrm{d}\boldsymbol{A}(t)}{\mathrm{d}t}\boldsymbol{B}(t) + \boldsymbol{A}(t)\frac{\mathrm{d}\boldsymbol{B}(t)}{\mathrm{d}t}
\]
4) 逆矩阵的微分
如果矩阵 \(\boldsymbol{A}(t)\) 及其逆矩阵 \(\boldsymbol{A}^{-1}(t)\) 对 \(t\) 是可微的,那么 \(\boldsymbol{A}^{-1}(t)\) 的微分可以导出如下等式:
\[
\frac{\mathrm{d}}{\mathrm{d}t}[\boldsymbol{A}(t)\boldsymbol{A}^{-1}(t)] = \frac{\mathrm{d}\boldsymbol{A}(t)}{\mathrm{d}t}\boldsymbol{A}^{-1}(t) + \boldsymbol{A}(t)\frac{\mathrm{d}\boldsymbol{A}^{-1}(t)}{\mathrm{d}t}
\]
因为
\[
\frac{\mathrm{d}}{\mathrm{d}t}[\boldsymbol{A}(t)\boldsymbol{A}^{-1}(t)] = \frac{\mathrm{d}}{\mathrm{d}t}\boldsymbol{I} = \boldsymbol{0}
\]
故
\[
\boldsymbol{A}(t)\frac{\mathrm{d}\boldsymbol{A}^{-1}(t)}{\mathrm{d}t} = -\frac{\mathrm{d}\boldsymbol{A}(t)}{\mathrm{d}t}\boldsymbol{A}^{-1}(t)
\]
于是有
\[
\frac{\mathrm{d}\boldsymbol{A}^{-1}(t)}{\mathrm{d}t} = -\boldsymbol{A}^{-1}(t)\frac{\mathrm{d}\boldsymbol{A}(t)}{\mathrm{d}t}\boldsymbol{A}^{-1}(t)
\]
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